Are we required to memorize the the structures of all 20 amino acids? Or is there an easier way?

No, please don't spend your time memorizing amino acid side chains. You will be given a table of amino acid structures as part of your exam, and they will even be categorized the same way we talked about them in class. You should, however, understand the consequences of being in one group versus another. What does it mean to be polar? What is the significance of being hydrophobic? And so on...